Graph colouring is hard for algorithms based on hilbert's nullstellensatz and gröbner bases

Forskningsoutput: Kapitel i bok/rapport/Conference proceedingKonferenspaper i proceeding

Abstract

We consider the graph k-colouring problem encoded as a set of polynomial equations in the standard way. We prove that there are bounded-degree graphs that do not have legal k-colourings but for which the polynomial calculus proof system defined in [Clegg et al. 1996, Alekhnovich et al. 2002] requires linear degree, and hence exponential size, to establish this fact. This implies a linear degree lower bound for any algorithms based on Gröbner bases solving graph k-colouring using this encoding. The same bound applies also for the algorithm studied in a sequence of papers [De Loera et al. 2008, 2009, 2011, 2015] based on Hilbert's Nullstellensatz proofs for a slightly different encoding, thus resolving an open problem mentioned, e.g., in [De Loera et al. 2009] and [Li et al. 2016]. We obtain our results by combining the polynomial calculus degree lower bound for functional pigeonhole principle (FPHP) formulas over bounded-degree bipartite graphs in [Miksa and Nordström 2015] with a reduction from FPHP to k-colouring derivable by polynomial calculus in constant degree.

Detaljer

Författare
Externa organisationer
  • Sapienza University of Rome
  • KTH Royal Institute of Technology
Forskningsområden

Ämnesklassifikation (UKÄ) – OBLIGATORISK

  • Datavetenskap (datalogi)

Nyckelord

Originalspråkengelska
Titel på värdpublikation32nd Computational Complexity Conference, CCC 2017
RedaktörerRyan O'Donnell
FörlagSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (elektroniskt)9783959770408
StatusPublished - 2017 jul 1
PublikationskategoriForskning
Peer review utfördJa
Externt publiceradJa
Evenemang32nd Computational Complexity Conference, CCC 2017 - Riga, Lettland
Varaktighet: 2017 jul 62017 jul 9

Publikationsserier

NamnLeibniz International Proceedings in Informatics, LIPIcs
FörlagSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Volym79
ISSN (tryckt)1868-8969

Konferens

Konferens32nd Computational Complexity Conference, CCC 2017
LandLettland
OrtRiga
Period2017/07/062017/07/09